In fan-beam computed tomography (CT) systems, an X-ray source projects a fan-shaped beam which is collimated to lie within an xy-plane of a Cartesian coordinate system termed the imaging plane. The X-ray beam passes through the object being imaged, such as a medical patient, and impinges upon a one-dimensional array of radiation detectors. Each detector produces an electrical signal that is a measurement of the attenuation of the X-ray beam by the object. The source and detector array in a conventional CT-system are rotated on a gantry within the imaging plane around the object through an angle of 180-360°, wherein a set of views made at different angular orientations during one revolution of the X-ray source and detector are acquired. From the acquired attenuation data, it is possible to reconstruct a 2D-image of the xy-plane through the object. A common way of performing the image reconstruction in fan-beam CT is for example filtered backprojection.
Image reconstruction becomes mathematically more complex when the X-ray beam is cone-shaped and the detector accordingly is a 2D-array of detector elements. However, these geometries are generally used in interventional or angiographic X-ray systems, for example in C-arm-systems where the source and the detector are mounted on the ends of a C-shaped arm which is adapted to rotate around the patient. On the other hand, it is highly desirable to be able to obtain an three-dimensional image of a volume within the patient by rotating the C-arm once around the patient (One rotation around e.g. 180-360°, during which a set of X-ray projections is acquired, is called 3D scan). Therefore, 3D-image reconstruction from circular cone-beam data has been an active research field for the last decades. A practical solution for circular-based source trajectories is disclosed in L. A. Feldkamp, L. C. Davis and J. W. Cress, “Practical cone beam algorithm” J.Opt. Soc. Am. A1, 612-619 (1984). The Feldkamp method is an analytic reconstruction approach. Unfortunately, analytic reconstruction approaches have to be adapted to each novel acquisition geometry. In other words, the Feldkamp method cannot readily be used with a non-circular source trajectory.
Recently, X-ray systems having another acquisition geometry have become available, namely where the X-ray source and the detector are mounted on telescopic arms. However, these systems have not been used to reconstruct 3D images, because in such an X-Ray system, the X-ray source would move along a planar polygon-based trajectory during a 3D scan. Unfortunately, no reconstruction algorithm is available for cone beam projections obtained with a non-circular trajectory.